Advertisement

An elitist non-dominated sorting bat algorithm NSBAT-II for multi-objective optimization of phthalic anhydride reactor

  • Shiv Prakash
  • Vibhu Trivedi
  • Manojkumar RamtekeEmail author
Original Article

Abstract

Nature inspired meta-heuristic algorithms are an integral part of modern optimization techniques. One such algorithm is bat algorithm which is inspired from echolocation behavior of bats and has been successfully applied to non-linear single-objective optimization problems. In this paper, a multi-objective extension of bat algorithm is proposed using the concepts of Pareto non-dominance and elitism. The novel algorithm is tested using thirty multi-objective test problems. The performance is measured using metrics namely, hyper-volume ratio, generational distance and spacing. The newly developed algorithm is then applied to a real-world multi-objective optimization problem of a phthalic anhydride reactor. It shows faster convergence for test problems as well as the industrial optimization problem than two popular nature inspired meta-heuristic algorithms, i.e. multi-objective non-dominated sorting particle swarm optimization and real-coded elitist non-dominated sorting genetic algorithm.

Keywords

Genetic algorithm Bat algorithm Particle swarm optimization Phthalic anhydride reactor 

List of symbols

Amin

Minimum loudness

Amax

Maximum loudness

Ai

Loudness of ith bat

At

Average loudness of all the bats at tth iteration

BA

Bat algorithm

e

Exponential

fi

Frequency of ith bat

fmin

Minimum frequency

fmax

Maximum frequency

Ng

Bats having rank 1

Np

Total number of solutions (bats) in the population

NV

Total number of decision variables

NSPSO

Non-dominated sorting PSO

NSBAT-II

Elitist non-dominated sorting BA

r

Pulse emission rate

ri

Pulse emission rate at ith bat

ri,t

Pulse emission rate at tth iteration

RNSGA-II

Real-coded NSGA-II

t

Number of iterations

tmax

User-specified maximum number of iterations

vi

Current velocity of ith bat

\(v_{i,j}^{0}\)

Initial ith bat velocity

\(v_{i,j}^{t}\)

jth component of ith bat velocity at tth iteration

wi

Inertia weight for ith bat

\(\varvec{x}_{i}\)

Current position of ith bat

\(x_{j}^{low}\)

Lower value of decision variable

\(x_{j}^{high}\)

Upper of decision variable

\(x_{g}^{best}\)

Total number of solution having rank 1

\(x_{i,j}^{0}\)

jth component of ith position at 0th iteration

\(x_{i,j}^{t}\)

jth component of ith position at tth iteration

\(\varepsilon\)

Random number between −1 and 1

Notes

Acknowledgments

The partial financial support from Science and Engineering Research Board, Department of Science and Technology, Government of India, New Delhi [through Grant SERB/F/1510/2014-2015, dated June 5, 2014] is gratefully acknowledged.

Supplementary material

13198_2016_467_MOESM1_ESM.docx (156 kb)
Supplementary material 1 (DOCX 156 kb)

References

  1. Aggarwal M, Hanmandlu M (2015) Representing uncertainty with information sets. IEEE Trans Fuzzy Syst 6706:1–15. doi: 10.1109/TFUZZ.2015.2417593 Google Scholar
  2. Ali M, Siarry P, Pant M (2011) An efficient differential evolution based algorithm for solving multi-objective optimization problems. Eur J Oper Res 217:404–416. doi: 10.1016/j.ejor.2011.09.025 MathSciNetzbMATHGoogle Scholar
  3. Bahmani-Firouzi B, Azizipanah-Abarghooee R (2014) Optimal sizing of battery energy storage for micro-grid operation management using a new improved bat algorithm. Int J Electr Power Energy Syst 56:42–54. doi: 10.1016/j.ijepes.2013.10.019 CrossRefGoogle Scholar
  4. Bansal JC, Sharma H, Jadon SS, Clerc M (2014) Spider monkey optimization algorithm for numerical optimization. Memet Comput 6(1):31–47. doi: 10.1007/s12293-013-0128-0 CrossRefGoogle Scholar
  5. Banzhaf W, Nordin P, Keller RE, Francone FD (2011) Genetic programming: an introduction, vol 31. Morgan Kaufmann, San Francisco. doi: 10.1109/5254.846288 zbMATHGoogle Scholar
  6. Bhat GR, Gupta SK (2008) MO optimization of phthalic anhydride industrial catalytic reactors using guided GA with the adapted jumping gene operator. Chem Eng Res Des 86:959–976. doi: 10.1016/j.cherd.2008.03.012 CrossRefGoogle Scholar
  7. Carmelo Filho JA, De Lima Neto FB, Lins AJ, Nascimento AI, Lima MP (2008) A novel search algorithm based on fish school behavior. In: Conference proceedings—IEEE international conference on systems, man and cybernetics, pp 2646–2651. doi: 10.1109/ICSMC.2008.4811695
  8. Černý V (1985) Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm. J Optim Theory Appl 45(1):41–51. doi: 10.1007/BF00940812 MathSciNetCrossRefzbMATHGoogle Scholar
  9. Chawla M, Duhan M (2015) Bat algorithm: a survey of the state-of-the-art. Appl Artif Intell 29(6):617–634. doi: 10.1080/08839514.2015.1038434 CrossRefGoogle Scholar
  10. Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197. doi: 10.1109/4235.996017 CrossRefGoogle Scholar
  11. Doğan B, Ölmez T (2015) A new metaheuristic for numerical function optimization: vortex search algorithm. Inf Sci 293:125–145. doi: 10.1016/j.ins.2014.08.053 CrossRefGoogle Scholar
  12. Dorigo M, Blum C (2005) Ant colony optimization theory: a survey. Theoret Comput Sci 344(2–3):243–278. doi: 10.1016/j.tcs.2005.05.020 MathSciNetCrossRefzbMATHGoogle Scholar
  13. Duan H, Luo Q (2015) New progresses in swarm intelligence–based computation. Int J Bio-Inspired Comput 7(1):26–35. http://www.inderscienceonline.com/doi/abs/10.1504/IJBIC.2015.067981
  14. Durillo JJ, Nebro AJ (2011) JMetal: a java framework for multi-objective optimization. Adv Eng Softw 42(10):760–771. doi: 10.1016/j.advengsoft.2011.05.014 CrossRefGoogle Scholar
  15. Eskandar H, Sadollah A, Bahreininejad A, Hamdi M (2012) Water cycle algorithm—a novel metaheuristic optimization method for solving constrained engineering optimization problems. Comput Struct 110–111:151–166. doi: 10.1016/j.compstruc.2012.07.010 CrossRefGoogle Scholar
  16. Fister I, Yang X-S, Fister I, Brest J, Fister D (2013) A brief review of nature-inspired algorithms for optimization. arXiv Preprint arXiv:1307.4186 80(3):1–7. http://arxiv.org/abs/1307.4186
  17. Fong S, Wang X, Xu Q, Wong R, Fiaidhi J, Mohammed S (2015) Recent advances in metaheuristic algorithms: does the makara dragon exist? J Supercomput. doi: 10.1007/s11227-015-1592-8
  18. Gandomi AH, Alavi AH (2012) Krill Herd: a new bio-inspired optimization algorithm. Commun Nonlinear Sci Numer Simul 17(12):4831–4845. doi: 10.1016/j.cnsns.2012.05.010 MathSciNetCrossRefzbMATHGoogle Scholar
  19. Gandomi AH, Yang XS, Alavi AH (2013) Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems. Eng Comput 29(1):17–35. doi: 10.1007/s00366-011-0241-y MathSciNetCrossRefGoogle Scholar
  20. Ghaemi M, Feizi-Derakhshi M-R (2014) Forest optimization algorithm. Expert Syst Appl 41(15):6676–6687. doi: 10.1016/j.eswa.2014.05.009 CrossRefGoogle Scholar
  21. Goel L, Gupta D, Panchal VK (2012) Hybrid bio-inspired techniques for land cover feature extraction: a remote sensing perspective. Appl Soft Comput J 12(2):832–849. doi: 10.1016/j.asoc.2011.10.006 CrossRefGoogle Scholar
  22. Goldberg David, Holland John (1988) Genetic algorithms and machine learning. Mach Learn 3:95–99. doi: 10.1023/A:1022602019183 CrossRefGoogle Scholar
  23. Huband S, Hingston P, Barone L, While L (2006) A review of multiobjective test problems and a scalable test problem toolkit. IEEE Trans Evol Comput 10(5):477–506. doi: 10.1109/TEVC.2005.861417 CrossRefzbMATHGoogle Scholar
  24. Jordehi AR (2015) Chaotic bat swarm optimisation (CBSO). Appl Soft Comput 26:523–530. doi: 10.1016/j.asoc.2014.10.010 CrossRefGoogle Scholar
  25. Jun L, Liheng L, Xianyi W (2015) A double-subpopulation variant of the bat algorithm. Appl Math Comput 263:361–377. doi: 10.1016/j.amc.2015.04.034 MathSciNetGoogle Scholar
  26. Karaboga D, Basturk B (2008) On the performance of artificial bee colony (ABC) algorithm. Appl Soft Comput 8(1):687–697. doi: 10.1016/j.asoc.2007.05.007 CrossRefGoogle Scholar
  27. Kaveh A, Farhoudi N (2013) A new optimization method: dolphin echolocation. Adv Eng Softw 59:53–70. doi: 10.1016/j.advengsoft.2013.03.004 CrossRefGoogle Scholar
  28. Kaveh A, Talatahari S (2010) A novel heuristic optimization method: charged system search. Acta Mech 213(3–4):267–289. doi: 10.1007/s00707-009-0270-4 CrossRefzbMATHGoogle Scholar
  29. Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Neural Networks, 1995. Proceedings of IEEE international conference on, vol 4, pp 1942–1948. doi: 10.1109/ICNN.1995.488968
  30. Kirkpatrick S, Gelatt CDD, Vecchi MP, Vecchi MP (1983) Optimization by simulated annealing. Science 220(4598):671–680. doi: 10.1126/science.220.4598.671 MathSciNetCrossRefzbMATHGoogle Scholar
  31. Krishnanand KN, Ghose D (2009) Glowworm swarm optimization for simultaneous capture of multiple local optima of multimodal functions. Swarm Intell 3(2):87–124. doi: 10.1007/s11721-008-0021-5 CrossRefGoogle Scholar
  32. Lee KS, Geem ZW (2005) A new meta-heuristic algorithm for continuous engineering optimization: harmony search theory and practice. Comput Methods Appl Mech Eng 194(36–38):3902–3933. doi: 10.1016/j.cma.2004.09.007 CrossRefzbMATHGoogle Scholar
  33. Lin K-C, Zhang K-Y, Huang Y-H, Hung JC, Yen N (2016) Feature selection based on an improved cat swarm optimization algorithm for big data classification. J Supercomput. doi: 10.1007/s11227-016-1631-0
  34. Mello RF, Andrade Filho JA, Senger LJ, Yang LT (2008) Grid job scheduling using route with genetic algorithm support. Telecommun Syst 38(3–4):147–160. doi: 10.1007/s11235-008-9101-5 CrossRefGoogle Scholar
  35. Meng XB, Gao XZ, Liu Y, Zhang H (2015) A novel bat algorithm with habitat selection and Doppler effect in echoes for optimization. Expert Syst Appl 42:6350–6364CrossRefGoogle Scholar
  36. Mirjalili S, Mirjalili SM, Yang XS (2013) Binary bat algorithm. Neural Comput Appl. doi: 10.1007/s00521-013-1525-5 Google Scholar
  37. Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61. doi: 10.1016/j.advengsoft.2013.12.007 CrossRefGoogle Scholar
  38. Niu J, Zhong W, Liang Y, Luo N, Qian F (2015) Fruit fly optimization algorithm based on differential evolution and its application on gasification process operation optimization. Knowl Based Syst 88:253–263. doi: 10.1016/j.knosys.2015.07.027 CrossRefGoogle Scholar
  39. Ramteke M, Ghune N, Trivedi V (2015) Simulated binary jumping gene: a step towards enhancing the performance of real-coded genetic algorithm. Inf Sci 325:429–454. doi: 10.1016/j.ins.2015.07.033 CrossRefGoogle Scholar
  40. Rashedi E, Nezamabadi-pour H, Saryazdi S (2009) GSA: a gravitational search algorithm. Inf Sci 179(13):2232–2248. doi: 10.1016/j.ins.2009.03.004 CrossRefzbMATHGoogle Scholar
  41. Sadollah A, Bahreininejad A, Eskandar H, Hamdi M (2013) Mine blast algorithm: a new population based algorithm for solving constrained engineering optimization problems. Appl Soft Comput J 13(5):2592–2612. doi: 10.1016/j.asoc.2012.11.026 CrossRefGoogle Scholar
  42. Schott JR (1995) Fault tolerant design using single and multicriteria genetic algorithm optimization. Massachusetts Institute of Technology, BostonGoogle Scholar
  43. Sedighizadeh M, Faramarzi H, Mahmoodi MM, Sarvi M (2014) Hybrid approach to FACTS devices allocation using multi-objective function with NSPSO and NSGA-II algorithms in fuzzy framework. Int J Electr Power Energy Syst 62:586–598. doi: 10.1016/j.ijepes.2014.04.058 CrossRefGoogle Scholar
  44. Storn R, Price K (1997) Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim. doi: 10.1023/A:1008202821328 MathSciNetzbMATHGoogle Scholar
  45. Van Veldhuizen DA, Lamont GB (1998) Multiobjective evolutionary algorithm research: a history and analysis. Technical Report TR-98-03, Department of Electrical and Computer Engineering, Graduate School of Engineering, Air Force Institute of Technology, Wright-Patterson AFB, OhioGoogle Scholar
  46. Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1(1):67–82. doi: 10.1109/4235.585893 CrossRefGoogle Scholar
  47. Wu TH, Chung SH, Chang CC (2010) A water flow-like algorithm for manufacturing cell formation problems. Eur J Oper Res 205(2):346–360. doi: 10.1016/j.ejor.2010.01.020 CrossRefzbMATHGoogle Scholar
  48. Xie J, Zhou Y, Chen H (2013) A novel bat algorithm based on differential operator and lévy flights trajectory. Comput Intell Neurosci 1–23Google Scholar
  49. Yang X (2011) Bat algorithm for multiobjective optimization. Int J Bio-Inspir Comput 3(5):267–274. doi: 10.1504/IJBIC.2011.042259 CrossRefGoogle Scholar
  50. Yang XS (2013) Multiobjective firefly algorithm for continuous optimization. Eng Comput 29(2):175–184. doi: 10.1007/s00366-012-0254-1 CrossRefGoogle Scholar
  51. Yang XS, He X (2013) Bat algorithm: literature review and applications. Int J Bio-Inspir Comput 5(3):141. doi: 10.1504/IJBIC.2013.055093 CrossRefGoogle Scholar
  52. Yang XS, Karamanoglu M, He X (2013) Multi-objective flower algorithm for optimization. Procedia Comput Sci 18:861–868. doi: 10.1016/j.procs.2013.05.251 CrossRefGoogle Scholar
  53. Yazdani M, Jolai F (2015) Lion optimization algorithm (LOA): a nature-inspired metaheuristic algorithm. J Comput Design Eng. doi: 10.1016/j.jcde.2015.06.003 Google Scholar
  54. Yilmaz S, Kucuksille EU (2013) Improved bat algorithm (IBA) on continuous optimization problems. Lect Notes Softw Eng 1(3):279–283. doi: 10.7763/LNSE.2013.V1.61 CrossRefGoogle Scholar
  55. Yılmaz S, Küçüksille EU (2015) A new modification approach on bat algorithm for solving optimization problems. Appl Soft Comput 28:259–275. doi: 10.1016/j.asoc.2014.11.029 CrossRefGoogle Scholar
  56. Yu JQ, Li VOK (2015) A social spider algorithm for global optimization. Appl Soft Comput 30:614–627. doi: 10.1016/j.asoc.2015.02.014 CrossRefGoogle Scholar
  57. Zhao W, Wang L (2016) An effective bacterial foraging optimizer for global optimization. Inf Sci 329:719–735. doi: 10.1016/j.ins.2015.10.001 CrossRefGoogle Scholar
  58. Zhou Y, Li L, Ma M (2015a) A complex-valued encoding bat algorithm for solving 0–1 knapsack problem. Neural Process Lett 1–24. doi: 10.1007/s11063-015-9465-y
  59. Zhou Y, Luo Q, Chen H, He A, Wu J (2015b) A discrete invasive weed optimization algorithm for solving traveling salesman problem. Neurocomputing 151:1227–1236. doi: 10.1016/j.neucom.2014.01.078 CrossRefGoogle Scholar
  60. Zitzler E, Thiele L (1998) Multiobjective optimization using evolutionary algorithms—a comparative case study. Parallel Probl Solv Nat 1498:292–301. doi: 10.1007/BFb0056872 Google Scholar

Copyright information

© The Society for Reliability Engineering, Quality and Operations Management (SREQOM), India and The Division of Operation and Maintenance, Lulea University of Technology, Sweden 2016

Authors and Affiliations

  • Shiv Prakash
    • 1
  • Vibhu Trivedi
    • 1
  • Manojkumar Ramteke
    • 1
    Email author
  1. 1.Department of Chemical EngineeringIndian Institute of Technology DelhiDelhiIndia

Personalised recommendations